Secondary Chern-Euler forms and the law of vector fields

Author:
Zhaohu Nie

Journal:
Proc. Amer. Math. Soc. **140** (2012), 1085-1096

MSC (2000):
Primary 57R20, 57R25

DOI:
https://doi.org/10.1090/S0002-9939-2011-11214-0

Published electronically:
July 1, 2011

MathSciNet review:
2869093

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Abstract: The Law of Vector Fields is a term coined by Gottlieb for a relative Poincaré-Hopf theorem. It was first proved by Morse and expresses the Euler characteristic of a manifold with boundary in terms of the indices of a generic vector field and the inner part of its tangential projection on the boundary. We give two elementary differential-geometric proofs of this topological theorem in which secondary Chern-Euler forms naturally play an essential role. In the first proof, the main point is to construct a chain away from some singularities. The second proof employs a study of the secondary Chern-Euler form on the boundary, which may be of independent interest. More precisely, we show by explicitly constructing a primitive that away from the outward and inward unit normal vectors, the secondary Chern-Euler form is exact up to a pullback form. In either case, Stokes' theorem is used to complete the proof.

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Additional Information

**Zhaohu Nie**

Affiliation:
Department of Mathematics, Penn State Altoona, 3000 Ivyside Park, Altoona, Pennsylvania 16601

Address at time of publication:
Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322

Email:
znie@psu.edu

DOI:
https://doi.org/10.1090/S0002-9939-2011-11214-0

Received by editor(s):
December 15, 2010

Published electronically:
July 1, 2011

Communicated by:
Jianguo Cao

Article copyright:
© Copyright 2011
American Mathematical Society